Optimizing Acquisition and Retention over Time
Adi Ditkowski
Department of Applied Mathematics,
Tel Aviv University, Tel Aviv, Israel 69978
Barak Libai
Recanati Graduate School of Business Administration,
Tel Aviv University, Tel Aviv, Israel 69978
Eitan Muller
Stern School of Business
New York University, New York, NY 10012
Recanati Graduate School of Business Administration,
Tel Aviv University, Tel Aviv, Israel 69978
November 2003
Revised August 2005
The authors would like to thank Renana Peres, Roni Shachar, and three anonymous reviewers for their helpful comments and suggestions. The late Dick Wittink was particularly helpful and supportive on an earlier version of this manuscript. This paper has been partially funded by the Israel Institute for Business Research.
Optimizing Acquisition and Retention over Time
Abstract
While making informed decisions regarding investments in customer retention and acquisition becomes a pressing managerial issue, formal models that enable generalizations on this topic are still scarce. In this study we examine what should be the optimal path over time that firms should follow in their acquisition and retention spending in a growing market, in both monopoly and competitive settings. To do so, we develop a continuous formulation for firms’ customer equity that takes into account future acquisition of new customers. We find that that the optimal path of investment in acquisition and retention is generally a declining one: Firms should invest more in the early stages of the market evolution and less later on. Specifically, the equilibrium acquisition path is always declining, while retention policy either includes an initial blitz followed by a gradual decline over time, or else it increases initially and then declines for the rest of the planning horizons.
1. Introduction
The shift of many firms away from the product to the customer as a unit of analysis has drawn considerable attention to the use of customer equity as a central tool for firms’ market analyses (Hogan, Lemon, and Rust 2002). Using the customer equity approach, customers are viewed as assets that create long-term value for the firm. Managers are thus encouraged to optimize the investments in these assets over time in order to maximize the long-term net cash flow from their customers (Rust, Lemon, and Zeithaml 2004).
An important decision of the firm in this respect is the magnitude and timing of its investments in customer acquisition and retention. The growing availability of customer data, coupled with increasing awareness regarding the long-term impact of successful acquisition and retention strategies, make these decisions a pressing managerial issue. For example, Morgan Stanley, aiming at shifting from a product-centric to a client-centric organization, has made it a goal “…to make customer acquisition and retention a more analytical and measurable process” (Lester 2003). Citibank has recently discontinued the position of “Head of Acquisition Marketing”, explaining this move with its shift in focus from customer acquisition to retention (Marketing 2003). In the telecommunications industry, European mobile operators are observed refocusing their business strategies on retaining—rather than acquiring—customers (Europemedia 2002). On the other hand, a survey among B2B managers suggests that it is customer acquisition, more than retention, that managers will focus on in 2005 (Maddox and Krol 2005).
These examples demonstrate the need for tools as well as intuitive processes that will guide managers when making their investments in customer acquisition and retention. However, given the importance of this subject, the academic literature in this area is still limited, as only recently have marketers begun to rigorously examine the relationship between customer acquisition and retention (Reinartz, Thomas, and Kumar 2005; Thomas 2001). Current approaches to this problem are geared more to helping a given company set its acquisition and retention budget at a specific point in time and given its specific situation (Blattberg and Deighton 1996), and somewhat less at managerial generalizations.
The issue we investigate in this article is the optimal path over time that firms should follow when managing their acquisition and retention spending in a growing market, in order to maximize their long-term customer equity. To do so, we present a continuous time formulation, both under a monopoly and in a competitive setting, to calculate the long-term customer equity of firms that takes into account acquisition of future customers. We use optimal control methods and calculus of variations (Kamien and Schwartz 1991) to understand how investments in retention and acquisition should behave over time if firms want to maximize their long-term equity. Our results suggest that the optimal path of investment in acquisition and retention is generally a declining one: Firms should invest more in the early stages of the market evolution and less later on. Specifically, the equilibrium acquisition path is always declining, while retention policy either includes an initial blitz followed by a gradual decline over time, or else a decline from early on.
2. Dynamic customer equity
Retention and acquisition spending
While historically much attention has been devoted in the marketing literature to offensive actions aimed at acquiring new customers, since the mid-1980s, increasing attention has been paid to defensive strategies, or actions focusing on retaining existing customers (Zeithaml 2000). In one of the first rigorous treatments of this issue, Fornell and Wernerfelt (1987) showed various complaint handling programs that marketers use to affect retention, and concluded that in general, more attention should be paid to defensive strategies. Their theoretical findings were later followed by a series of papers by both practitioners and academics that pointed to the need to focus on retention, given its effect on the long-term bottom line (e.g., see Zeithaml 2000; Reichheld and Sasser 1990). Given its effect on the valuation of customers, the retention rate has also been suggested as an important factor affecting the value of firms in general (Gupta, Lehmann, and Stuart 2004).
As the focus on retention spending has become an important managerial issue and the basis for many of the investments in CRM systems, recent doubts have been raised as to whether this focus has gone too far, coming at the expense of profits and the growth of the company through new customers (Ambler 2001; Reinartz and Kumar 2000). Models and generalization on the acquisition / retention question might help to clear the picture on that point. However, while the long-run optimal balance of acquisition and retention budget allocation is recognized as one of the challenging tasks facing marketing resource allocation in general (Hanssens 2003), there is scant formal analysis in the marketing literature of these issues.
In a pioneering exception, Blattberg and Deighton (1996), followed by Berger and Nasr-Bechwati (2001) and Pfeifer (2005), suggested a simple deterministic managerial tool via which a firm can aim to optimize retention and acquisition spending in a static structure. Thomas (2001) drew marketers’ attention to the possible connection between acquisition and retention and the biases that it can create when analyzing customer retention without taking this connection into account. Verhoef and Donkers (2005) have demonstrated empirically that channels of customer acquisition may influence the retention rate.
Recently, Reinartz, Thomas, and Kumar (2005) presented one of the first rigorous statistical attempts to draw acquisition and retention inferences given customer-level data. They utilized data on various interactions with potential and current customers coming from a large multinational software and hardware supplier, and used a two-stage least square Probit model to estimate how changes in specific acquisition and retention activities would affect the firm’s profitability. Based on this case, they generalized regarding acquisition / retention strategies, for example, that suboptimal allocation of retention expenditures would have a higher impact on profitability compared with a suboptimal allocation of acquisition resources.
The question we ask in this article differs from the acquisition / retention balance in a static framework such as that described by Blattberg and Deighton (1996), or the more statistic-based approach given specific customer data of Reinartz, Thomas, and Kumar (2005). First we take the question of balancing acquisition and retention to the case of a growing market, in which acquisition efforts can affect the rate of market growth. Second, our research is not geared toward a specific company’s case, but rather toward the aim of generalizing as far as possible regarding the optimal path of retention and acquisition spending over time. The firm’s measure of profit to be optimized is customer equity, or the long-range profitability of a firm’s customer, which will be discussed next.
Customer equity in a growing market
Customer Lifetime Value (CLV) has gained increasing interest in recent years as a basic tool to help firms decide on the magnitude and the nature of investment in their customer relationships (Rust, Lemon, and Zeithaml 2004; Jain and Singh 2002; Blattberg, Getz, and Thomas, 2001; Reinartz and Kumar 2000). CLV represents the discounted cash flow a firm expects to receive from an individual customer over some extended period. For example, if the retention rate is r, the average profit from a customer is p, and the discount rate is i, then the CLV over a long time horizon (as t approaches infinity) is: (1)
This well-known formulation may be slightly modified based on specific assumptions regarding the exact time that the cash flow is received during each period, and the time during a given period when a defection is assumed to have occurred (Gupta and Lehmann 2003).
Note that the above formula assumes that when customers leave the firm, they do not come back (or if they come back, they are considered new customers). This “lost for good” assumption, which enables a relatively straightforward modeling of CLV, may be less robust in markets where consumers switch often among brands, such as with frequently purchased goods, and in such cases may result in the underestimation of the actual CLV. In such cases “migration models”, which utilize a Markov chain analysis may be a better fit (Rust, Lemon, and Zeithaml 2004). We examine a migration-type defection when analyzing the case of a duopoly.
Equation (1) represents the lifetime value of a single individual. When maximizing their long-range profit, firms will be more interested in customer equity, or the sum of the lifetime value of all of the firm’s customers (Rust, Lemon, and Zeithaml 2004; Venkatesan and Kumar 2004). While most applications have examined customer equity in the context of the value of the current customer base (Blattberg, Getz, and Thomas, 2001; Blattberg and Deighton 1996), Rust, Lemon, and Zeithaml (2004) suggest that customer equity should also include the value of future customers. Indeed, recent use of customer equity for firm valuation took into account the acquisition of new customers in a growing market (Gupta, Lehmann, and Stewart 2004).
The modeling of customer equity in a growing market is clearly of importance given the central role of new products in the sales of many firms. The importance of such cases is especially visible where customer equity models are used to calculate the value of firms, especially service firms such as electronic commerce retailers for whom customer relations are recognized as major assets (Gupta and Lehmann 2003). For example, it has been recently suggested that customer equity measures should play a much more central role in decisions regarding firms’ mergers and acquisitions (Selden and Colvin 2003). Therefore, since the firm’s value depends on future cash flow from its customers, all customers should be included in the calculations, including anticipated new customers. To differentiate this approach from customer equity measures that are based on the current customer base, we label the customer equity in a growing market as Dynamic Customer Equity (DCE).
Dynamic Customer Equity (DCE) model
This basic formulation of the Dynamic Customer Equity equation is similar in nature to the one presented in Gupta, Lehmann, and Stuart (2004), and therefore only briefly presented here. Historically, most formulations of the “lost for good” CLV analysis have been discrete. However, the derivation of the continuous analogy to the discrete case is essential for the maximization problem formulated next, and so expanded upon in this section.
Starting with the individual CLV, let x(t) be the state variable that denotes the probability that a consumer remains an active customer of the firm at time t. The discrete time formulation implies that and thus. It therefore follows that the first differences are given by . The continuous time analog is achieved by replacing the discrete differences by continuous differentiation:
(2)
The solution of equation (2) (for x(0) = 1) is given by:
It follows that the replication of Equation (1) for the continuous time formulation is given by:
(3)
Assume that the lifetime value of a single customer is as shown in (3). Since the firm acquires more customers with time, the cumulative number of customers acquired by the firm grows with time (customers may of course leave after they are acquired, depending on the retention rate). Since our calculations start in time zero, the sum of the lifetime value of each group of customers acquired at time t should be discounted to time zero. Let a(t) be the cumulative number of acquired customers up to time t. Hence, the number of consumers newly acquired at time t is da/dt. If we assume a general growth function of da/dt = f(t) where f(t) is a general continuous growth function, then the general form of the Dynamic Customer Equity is given by Equation (4), which is similar to the one derived in Gupta, Lehmann, and Stuart (2004).
(4)
We will start with the case of a monopoly that operates in a market whose conditions were described above. Monopolistic models are especially useful in understanding category-level effects, and are also relevant in the early market growth period when there are few competitors if any (Kalish 1983). We will consider the competitive case in the Section 4.
3. Dynamic optimal acquisition and retention of a monopolist
We now turn to a dynamic setting in which the firm can use its retention and acquisition budgets to control the growth and maintenance of its customer base. In the context of our model, retention spending will affect the retention rate r that becomes r(t), while acquisition spending affects the growth rate of the number of new customers, represented by g(t). Under our formulation, a change in retention spending affects the retention rate of all customers—new and old. Note that x(t), which was the percentage of active customers for a given cohort in (2) becomes in the dynamic model the percentage of customers out of the total customers acquired to date a(t) (i.e., of all cohorts) that are still active customers of the firm. Thus the firm can affect the rate of acquisition of new customers through its acquisition efforts, and the rate of retention through its retention efforts.
Costs can take on two forms: either total costs, which are independent of the number of customers, or else cost per-customer. For acquisition costs, the difference between total and per-customer is a choice of convenience only. If we denote the total acquisition costs by K(g), and the acquisition cost per potential customer by k(g), then the relationship between the two is given by the following: , where a(t) is the cumulative number of acquired customers up to period t, and m is the market potential. Consistent with Blattberg and Deighton (1996), we assume the cost functions to be monotonic and convex, i.e., and that, and similarly for k(g).
One should note that assuming a convex cost function is equivalent to assuming a concave effectiveness function. The model given in Blattberg and Deighton (1996) is the concave effectiveness retention function , where c is the dollar cost. This function is equivalent to the convex cost function, where r is the retention resulting from expenditures of $c. Similarly, the concave retention function is equivalent to the quadratic convex function .
In the retention costs case, the difference between total costs and per-customer cost is much more pronounced and case-dependent. Since the retention rate r is bounded between 0 and 1, the question is whether it depends on the number of consumers, or on the market potential alone. For example, for many Web-based firms, retention and the resultant repeat purchasing depends on factors such as categorizing users by their technological sophistication, ensuring perceived security, empowering users, and creating trust and commitment (Vatanasombut, Stylianou, and Igbaria 2004), none of which depend on the number of users. In contrast, the retention efforts of many brick-and-mortar firms such as call centers do depend on the number of users. We thus separate our analysis into these types of costs. As we demonstrate, the results are inherently similar, though not all results could be replicated for the two cases.
As regarding the acquisition costs, we assume monotonic convex retention costs, both total and per-customer, i.e.,and that , and similarly for c(r). In the following analysis, we will use both a general convex function for analytical proofs and a quadratic form function for the numerical analysis, commonly used in economics and marketing (see for example Kamien and Schwartz 1991).